A Jewish-Italian Hannakristmas

I'm in Cleveland at my dad's house for the annual event... the homemade gnocchi and sauce were great, the kids got lots of noisy plastic crap to forget about by tomorrow, and the vegan chocolate-peanut butter cake from Mustard Seed cafe was tasty: I had two, ok three, slices.

After desert, board games, and being shot at with nerf guns (which have gotten scarily high-caliber), I was talking with my 10-year-old half-brother about his school. Specifically, I wanted to see how he is doing in math. He is (not surprisingly) unable to articulate exactly what he is doing in math, so I was asking him specific questions to see what his math is like. Last year, I was surprised to find out he knew square roots already - I explained about cube and higher roots, and he picked it up instantly.

I wanted to see what he knew about fractions as a 4th grader at a typical Cleveland-area public school. I asked him if 1/2 or 3/4 was bigger.. way too easy. I asked him if 2/3 or 3/4 was bigger. He got it right, and quickly, but couldn't really explain why. I then asked him if 3/7 or 3/8 was bigger, and he said 3/7 immediately. I asked him to explain how he knew, and he looked at me like I was stupid, saying, "a seventh is bigger than an eighth, so..". I asked if he had worked with mixed numbers, and he hadn't, so I asked him to figure out what 3 1/4 - 1 1/2 is. He couldn't do it in his head, so I told him to get paper and draw a picture. That's all I said - he drew fractions circles correctly, crossed off a whole, the fourth, and then another fourth from a whole, and came up with 1 3/4.

I've only ever taught at DCP, so I don't have much of a frame of reference for knowing if he is above average or not, but this is the kind of thinking that all students must have to be successful in high school math. This is the kind of numeracy ability I want my students to develop; this way, when they get to a new problem, instead of giving up, they can reason it through and at least make progress. I struggle daily to get them to pay attention, to care, to think, to not give up when a problem is hard, and their mathematical progress is painfully slow. In a couple weeks, when we start reviewing for finals, and half the kids don't even remember what an integer is, it will be painful. But I know that, by the end of the year, most of my students will have improved their math abilities in many ways. Never as much as I want, but it will have to do! I just gave our grade-level equivalency test before break, and the median score has improved by 1.1 grade levels (from 5.9 to 7.0) and the average by 1.65 grade levels (from 5.76 to 7.42) since the summer. If I can squeeze that kind of growth or better out of them during the second semester, most will be in pretty good shape for next year.

A case study: freshmen's ability to listen...

(Background: cell phones are not permitted. If seen/heard, they are confiscated until parents pick them up.)

{Phone rings; I answer}

Me: C, they need you at the front desk with your cell phone, because your sister got hurt and they don't have your mom's cell phone number.

C: (Looking angry) But I don't even have a cell phone!

Me: They don't want to take your phone, they just need your mom's number.

C: But I don't have a phone!

Me: Just go...

(5 minutes later)

C: Mr. Greene, do you know why they wanted me? My sister hurt her leg, and they needed my mom's number!

Me: That's exactly what I told you.

C: You did?

More decimal and percent work

For the next lesson in Numeracy, I wanted to keep building students' ideas about what percents and decimals are, and how they relate to fractions. I've mentioned before that I am teaching the students to use bar modeling to solve word problems, but I haven't been posting the problems, and some examples would probably be nice. This week, I've started incorporating percents into the problems, which is, of course, throwing the students for a loop. But, they will get it eventually (and some already have), and I think that continually reinforcing the visual connection between percents and fractions is important. So here are the three problems I'm using this week:

Lesson 1

Diego and Dora both took a test in Algebra 1. Diego got 70% of the questions correct, which was 42 points. Dora did very well, and even got the bonus problem right, so she got a 105% on the test. How many points did Dora score on the test?

Lesson 2

By the end of tutorial, Mariana completed 45% of her homework. She spent 54 minutes working (the rest of the time, she was giggling with Gricelda). If she works at the same speed at home, how much longer will it take her to finish all of her homework?

Lesson 3

At the school dance, 70% of the students were girls, and the rest were boys. Ms. Vasquez wondered why there were so few boys there – she counted only 36 boys. How many students were at the dance in total?

We only do one problem like this per lesson (3 lessons per week - block schedule), because it really takes 15 - 20 minutes for the whole process (more when the students are unfocused) to play out. What's nice about these problems and this method is that it naturally connects percents to the work students have been doing for months drawing whole number bars (first) and then fraction bars. Right now, lots of students are still struggling, but I think it's more due to the proximity of vacation affecting their ability to care about math than a conceptual problem. We'll pick up with this after break as we review for finals, and I think it'll go better.

For this lesson, after finishing the problem solving portion, we did another class activity. I gave each group a set of 10 post-it notes with various decimals and percents, all between a pair of consecutive whole numbers. They had to stick their post-it notes to the board (where I had blue-taped up a long number line) drawing arrows with marker to indicate more precisely where the number should go. This only took about 5 minutes or so, and then I had everyone sit back down so we could evaluate how we did. I told them that they would earn 2 team points (whoopie!) for each number in the right place, and 1 bonus point (what can I say? Freshmen love their points!) if they could find a mistake in another team's positioning. We went through team by team, and I asked the class to point out any mistakes. The mistakes that were pointed out lead to additional discussions and modeling, until students seemed satisfied that everything was in the right place.

After doing this with my first period, I wasn't sure if the activity had been all that useful. I asked students if they found it useful (many did) and to share something that they had learned. This lead to some good questions and observations - the key one being that 2.45 is less than 2.5 (I still had the papers on the board from the previous lesson to refer back to) because of the value of each number; similarly, several realized that 2.45 is not more than 2.5 even though it has more numbers in it. Other students said that it helped them see better how a percent and a decimal could both be plotted together on the same line (yes!). So this made me feel better about the whole thing, and I didn't modify the lesson for the other periods.

Now that we've used manipulatives and done a couple of class activities, in tomorrow's lesson, students will be doing a worksheet I put together to get some solid independent practice time. They will start by drawing paper pieces (as in the last lesson) to represent a variety of decimals, fractions, and percents. Then, given a number in one form, they will have to write it in the other two forms. Finally, they have a bunch of problems where they must compare a pair of numbers - written in any form - to see which is larger.

The decimal point's job... where's the one?

This has been a long and stressful week, but it ended well yesterday. The biology classes were on a field trip, so my Algebra 2 class only had about 5 kids in it. I didn't bother trying to teach anything new - I just helped them with the homework, and then we spent the rest of the time looking at different problems they needed help with, and talking about stuff that was on their minds... some "remember when", since I taught most of them as Freshmen - they remember amazingly well things that I said over two years ago, as long as they are not math related, apparently... we talked through some of their fears about going to college... we reviewed fractions... I wish there were more times available to just sit and chat with students.

In my Numeracy classes, the goal of the day's lesson was to learn how the base-10 positional system works, with a focus on the decimal side of things. Specifically, I wanted them to be able to represent fractions and percents as decimals, and to understand how each decimal place relates to a specific base-10 fraction. This lesson went very poorly on Thursday with my first two classes, due to management issues (i.e. they were behaving terribly). Is there ever a point in time when you can get through a week without feeling like a first-year teacher again? Anyway, I didn't rewrite the lesson for my Friday classes, because I was confident that it was actually a good lesson, and worthwhile.. I just tightened it up, and made sure to stay on top of things behaviorally, and it really paid off. The discussions in both periods (and these are my two weaker classes) were quite rich and productive. On my board, I had taped up papers as in the graphic above, and told students that each smaller piece had been made by cutting the previous one up into 10 pieces. I asked how big each was, and they easily saw that each paper was 1/10 of the one to the left. I then asked which paper represented one whole. Some students thought it was the 10 big pieces, some thought it was the single full-size piece, some thought it was one of the smaller ones. They discussed it, and as I kept asking more students what they thought, in each class, several students started saying that any one of the pieces could be one whole. They convinced each other without much of my prompting - it was like magic! I then pulled out my magic Decimal Point cutout, and told them that the decimal point has one and only one job - to determine which shape is worth one whole. I taped the point up on the board (making the full sheet of paper the one whole for this lesson), and we then proceeded to label all the place values relative to one whole. All of a sudden, why the tenths are called "tenths" started to make sense for some students.

I showed what the fraction 71/100 would look like by taping up 7 strips and 1 tiny square in the right columns, and then translated this to the decimal 0.71. We did a couple more examples (I called students up to try some), and then they began asking about adding zeros at the end. Some thought it was ok, and others thought it would change the value. So I had them debate for a while if 0.6 and 0.60 were the same or not. Eventually, they were convinced that they were the same. So, to push them, I asked why 6 and 60 are not the same, which allowed me to show them that the 0's only change the value when they push a digit into a different place value. For each example, I also went back to the definition of percent (a fraction out of 100) to show that the tiny squares must therefore be considered 1% each. This made it easy to see why 0.36 would be the same as 36%, as they could see the 36 little squares on the board (comprised of 3 strips and 6 squares). It also made it easy to explain why 0.237 is the same as 23.7% - you have 23 little squares, and 7/10 of a little square.

I saw quite a few light bulb moments over the course of this lesson. I need to find a way to get my first two periods back on board now... We'll review this on Monday; I hope that, by the end of next week, students will all be able to compare and order decimals, and easily move back and forth between decimal, fraction, and percent representations. That would be quite an accomplishment, and a good way to go into break.

Fidelity in Math

In Algebra 2, the topic today was an overview of functions. Some students were having difficulty understanding the "each input has exactly one output" condition, and the previous example (percent score --> letter grade) just wasn't cutting it.

The follow-up example was much better. The domain was a set of boys' names, the range was a set of girls names, and the mapping was "dating". And, of course, one of the boys had an arrow pointing to three different girls. We discussed why this was not a function, and one student said, "So to be a function, they have to be faithful!". Exactly! I took her up on this, and had them add to their notes: Functions are Faithful! This instantly made sense to them, and this language carried forth through the rest of the lesson. I then added another boy pointing to one of the girls that was already in the list, and asked if everyone was still faithful. They said no, and we clarified things; our new "taken-as-shared" idea was that only the boys (the inputs) have to be faithful for it to be a function. (I mentioned that if all the girls were faithful too, then it is called a one-to-one function, and we'd look at that later.)

It was really amazing - even when we did examples involving decontextualized numbers, they were still very comfortable using the analogy: i.e., that set of ordered pairs is not a function because the 4 is being unfaithful! It even made the vertical line test a breeze to teach.

It's always nice to find something new to add to the bag of tricks.

Happy Tofurkey Day

If you've read my blog in the past, you may have noticed I haven't been writing much lately. I've had some bad health problems over the past few months, which have made it difficult for me to get anything but the essentials done. It's finally under control, and I am feeling relatively human again. I am thankful for strong drugs and medical advances!

So, my lack of energy along with the typical November doldrums have made my freshman Numeracy classes less effective and positive than I'd like. I hope that this mini-break will give us all a chance to recover a bit, and come back to end the semester strong. Here are some of the major things I need to work on:

1) Multiplication Tables.
I decided not to focus on teaching the tables this year to the whole class, as it is a waste of time to the 1/2 to 2/3 of the kids that know them. And, I'm not sure how to do it really effectively for those that are in high school and still don't know them. ALEKS doesn't deal with multiplication tables.

I've given students who need them 12 x 12 tables to keep in their binders and look at as they do things like reducing fractions. I need to find a way that they can actually work on improving, and I think this will be different for each student. Some combination of flash cards, games, incentives, and quizzes will be needed. But how to work this seamlessly into the class? Hmm... Needs more thought..

2) Classroom Culture / Readiness
I've posted about this in the past, and my readiness checker was working very well. But, recently, class hasn't been getting started very efficiently. The students recognize this, and freely admit that, now that they are well into school and have made friends, there is a lot more temptation to hang out between classes and avoid getting into their classroom or seat in the room until the last possible second.

I don't sweat it if we miss out on the first minute or two of class, but more seriously and annoyingly, school supplies seem to be growing ever more sparse (kids don't have pencils or binder paper, and act shocked when I ask them to take these things out!). It's like they buy some new stuff in September, and when they're out, that's it for the year. Also, binders have gotten to be a mess (i.e. holy terror) again - and that's for the kids who still have binders.

I need to come up with an incentive system to get this all back on track - and a way that forces me to stay on top of it.

3) Problem Sets - Singapore Bar Models
I overshot greatly with my initial stabs at assigning students problem sets. My goal was to get them started on longer-term planning, while assessing their ability to use the bar model method to solve word problems. My initial idea was to assign ten problems on Monday, due the following Monday. I would grade and return them, and they would have till the following Monday to do revisions to increase their scores. Sounds reasonable? Well, it was still way too much for them to handle. The second time around, I made mandatory progress checks during the week, to help keep them on track. I got more turned in the second time around, but they were still not very good. And, there is a big problem with cheating. Unfortunately, students don't really understand all the time what cheating is, and it's hard to get them to see it. I want them to work together to help each other; many of them think that copying someone's answers who is "helping you" isn't cheating. This is going to require a lot more coaching. Here are my initial ideas for how to run problem set #3:
- Reduce the number of problems from 10 to 6.
- Continue with the progress checks, for regular homework credit.
- The day the assignment is due, give a one- or two-question quiz with selected problems from the assignment, but with numbers changed. This will help me see which kids actually understand the work, and which kids copied.

Don't worry, I'm not dead...

My Numeracy students are now about a month into their ALEKS experience. I started all students out on the third grade standards level (the lowest ALEKS goes), and on average, my students scored around 50% mastery on their initial diagnostics. At this point, some of the students have completed level 3 and are onto level 4, and many others are close to completing the level. There are the stragglers too, of course. I'll do more detailed stats later on. The goal I've set with students is that they should try to complete 3 entire levels by the end of the year (i.e. 3 years of growth in math ability). I was skeptical at first, but after seeing how the students interact with the program, I have much more hope. ALEKS is not a creative, fun, snazzy program. Essentially, students get a sample problem to try. If they don't know how to do it, they read an explanation and try again. When they get a certain type of question right 3 or 4 times in a row, without asking for help, the concept is added to their pie chart. Periodically, they are re-assessed by the program, and concepts they no longer know are pulled back out of their pie chart.

I have been impressed by how self-reliant the students are being. They are managing to read the explanations and figure out the problems on their own. Some students are really getting into it, and are bragging to each other about how much of their pie they have completed. They have also figured out that getting a problem wrong, or clicking on the "explain" button causes the program to require more correct problems to add the concept to the pie. For that reason, they are actually trying harder to get the problem right the first time. The immediate feedback has been very helpful for the students. My favorite moments are now at the end of class; sometimes, when I tell students they need to log off, a few will be like "oh wait, let me just get this one last problem so I can add it to my pie".

Right now, I am just assessing them on time spent on ALEKS - not on the actual amount of progress being made; it seems to be effective enough, and the whole point is to allow students to work at their own pace. We'll see if I need to modify that policy in the future.

On a different note, we have been working on bar modeling to solve word problems every class for 15-20 minutes. I assigned the first problem set as homework last week, and I graded them this weekend. They were quite bad. It's always a bad feeling when you realize your students are a lot farther behind than you thought. I've pushed ahead into more complicated problems, but I just realized that many students are still having trouble with the basics. That's ok.. we'll just cycle back to the beginning and have another go at it.

In Algebra 2, we've started in with the basic idea of logarithms, using the Big L notation I wrote about in an earlier post. I think it is working well. We have been focusing on the similarities between roots and logs: in a root, the index tells you the exponent, and you are looking for the base. In a log, the subscript tells you the base, and you are looking for the exponent. Last year, many students had trouble in power expressions determining when to use a log or a root; I think they will have a much better understanding of it this year.

Our Department Picture: Math Rox!

Are you ready for this?

I quickly mentioned the Readiness Checker idea a few posts ago, as a new idea I got from another teacher. I want to revisit it here, now that school has been under way for a few weeks. I have to say, it is simply excellent. In the past, I've done a variety of readiness checks, with various degrees of success. This is working better than any other method I've tried.

So here's how it works. To be ready, students must, *by the time the bell rings* do the following:

• Take out binder


• Take out homework (and Readiness Checker)


• Have a pencil sharp and ready


• Put backpack in back of the room


• Begin working on the Do Now

If all of those things are completed, the student earns a sticker or stamp on their Readiness Checker. When the checker is filled in (I currently have 9 spaces on mine, but will probably extend it to 12 for the next round), it turns into a "get out of homework" pass. This has the benefit of putting a nice positive incentive on being ready for class, no negative consequence, and it is not directly tied to points in the grade.

It has been working like a charm in my 9th grade Numeracy classes. Most of the time, I have at least 3/4 of the students earn a sticker (often more), which lets class start quietly, focused, and on track. Of course, there are off days (like Friday afternoons), but overall this has been a fantastic new innovation. If you teach students that have difficulty getting started, I highly recommend a system like this. A couple of our 12th grade English teachers are doing this too (I was surprised, but they say the students love it.) Best of all, there is no added management on your part - if the student loses their checker, give them a nice new blank one. (So far, though homework sometimes "gets lost", I haven't had a single student lose their Readiness Checker. What a surprise! :)

What we have to watch out for (a partial list)

San Jose, though a relatively safe city overall, does have a significant gang problem. Our students are generally not involved in gangs directly (though we do get the occasional hard core kid), but their communities are infused in a wash of red and blue, and gang symbols are everywhere. We work hard to keep this out of our school, so that all our kids can be and feel safe, and part of that means clamping down on the little behaviors that can flare up into big problems. So, aside from all of the normal things a teacher needs to watch out for, here are some others, any of which will get a kid put on a strict "gang contract" (which usually means that further behavior will end them up in a discipline committee meeting to discuss their behavior and their desire to remain at DCP).

• Red or blue markings on clothes or shoes


• Red or blue hair rubber bands, red or blue pens sticking out of pockets or used to hold up hair, red or blue nail polish and makeup


• Students writing in red pen (blue is too ubiquitous to try to prevent)


• Crossing out 3s or 4s; replacing "e"s with "3"s or writing "e"s backwards


• Using the numbers 3, 13, 4, or 14 inappropriately


• Showing problems with colors (i.e. a student given a blue whiteboard marker to write with who refuses and trades for a red)


• Certain tags like Sur, Norte, 408, ESSJ, Sharks


• Markings at the base of the thumb (3 or 4 dots)


• Roman numerals XIII or XIV, and clever ways to write them, such as dotting your "i"s with an "x", like in the word "live"


• Certain hand gestures


• The UFW eagle has also become a gang symbol. I had a couple of students building it out of unit cubes last week!

There are more, but that's all I can think of off the top of my head. It may seem trivial or ridiculous to watch out for these things, but it is amazing how serious they can be, and how they can lead directly to students getting physically intimidated or hurt.